Questions tagged [forcing]
Forcing is a set theoretic method used mainly for proving independence results. For questions about forcing function please use (differential-equations).
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Is the dominating number of this continuum graph a small cardinal?
Define the relation $\sim$ over $\mathcal C := \{0, 1\}^{\mathbb N}$ given by $(x_n)_n\sim (y_n)_n$ iff $\exists^\infty k: a_{k+i} = b_{k+i},~i=0, 1, \dots, k-1$.
What is the least size of a subset $\...
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Why take separative quotient in forcing?
In many expositions of topics on forcing, the poset is assumed to be separative, otherwise we pass to the separative quotient. For example, this is the convention taken in this introduction to forcing ...
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An exercise regarding Martin's Axiom
Define $\mathbb{P}$ = {$\left<s,F\right>:s\in\omega^{<\omega}\wedge F\subseteq\omega^\omega\wedge|F|<\aleph_0$}. Define $\left<t,G\right>\leq\left<s,F\right>$ if and only if
(1)...
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$\Delta$-system Lemma to disprove existence of an uncountable antichain?
I am trying answer a question:
Let $\mathbb{P} = \{p \mid p \text{ is a finite function } \omega_2 \times \omega \to 2\}$ ordered by revesre inclusion, i.e. $p<q$ iff $p \supseteq q$. Does $\...
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Examples of first-order claims about the reals that are not preserved under forcing
I am looking for an example of a first-order sentence in the signature of the real numbers, $(+,\times, <, 0,1)$, that is true when translated in the language of set theory in the natural way, but ...
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Ultrafilters used in Boolean-Valued Forcing
I am beginning to read a paper called "Well-Founded Boolean Ultrapowers as Large Cardinal Embeddings" by Joel David Hamkins and Daniel Evan Seabold. In the first section, they review the ...
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Why can we not use Cohen forcing to force the continuum to be $\aleph_{\omega}$?
I've gone through the constructions to change add many subsets to different cardinals, and know that Easton's theorem says that the power function can consistently be anything not inconsistent with ...
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Proof that P-Names allow the generation of all sets derivable from G in M[G]
Kunen says about Forcing : The first step is to define M [G]. Roughly, this will be the set of all sets which can be constructed from G by applying set-theoretic processes definable
in M. Each element ...
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Explanation of last step in proof of Lemma 4.2 in paper "An Axiomatic Approach to Forcing in a General Setting"
The paper "An Axiomatic Approach to Forcing in a General Setting" by R. Freire and P Holy includes the proof of Lemma 4.2.
(Note that in the paper a 'generic filter' means a filter in a ...
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Necessity of Axiom of Choice for unordered pairs of real subsets.
Find a choice function on $\{\{X,Y\} \vert X,Y \in \mathcal{P}(\mathbb{R})\}$
While reading Adrien Douady's book "Algèbre et Théorie Galoisienne", the first chapter focuses on the axiom of ...
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Linear ordering of reals in Cohen's first model
The presentation of Cohen's first model that I'm most familiar with is to start with a forcing extension by $Add(\omega,\omega)$, consider the group of automorphisms of $Add(\omega,\omega)$ that ...
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Forcing to add a $\square_\lambda$-sequence is $\mathrm{<}\lambda^+$-strategically closed
For an uncountable cardinal $\lambda,$ a $\square_\lambda$-sequence is a sequence $(C_\alpha: \alpha\in \lim(\lambda^+))$ such that
Each $C_\alpha$ is a club in $\alpha$ with order type $\le \lambda.$...
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Homogeneity of Lévy Collapse
I'm reading the section from Jech's Set Theory regarding the Lévy Collapse $Coll(\aleph_0,<\lambda)$. The following is a lemma towards proving the homogeneity of the Lévy Collapse:
Here Jech is ...
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Questions on Jech's proof of the independence of AC from the ordering principle
In the the book The Axiom of Choice section 5.5, Jech presents a proof of the independence of the axiom of choice from the ordering principle (every set can be linearly ordered). There are some ...
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Master Conditions Clarification
I'm having some trouble parsing the following passage from the Handbook of Set Theory:
I'm having trouble checking the statement that is "routine to check." My thought is that since $D,\...
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Definition of Separately Generic Forcing Extensions
This is a very simple question, but I'm having trouble finding a definition for separately generic forcing extensions. I know that if $G$ and $H$ are mutually $P-$generic over some ground model $M$, ...
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Quotient forcing and intermediate model
I previously asked a question about the factor lemma in Jech, and from the comments I realised that there is something earlier that I need to understand.
Let $Q$ be some forcing notion in $M$ and let $...
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Is Forcing only possible in first-order theories?
I'm trying to understand forcing in set theory. From what I understood it necessary that the Löwenheim-Skolem theorem holds to extend models or to force extensions. So, is forcing only possible in ...
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The Factor Lemma in Levy collapse
I'm following Jech and in page 518 there is the following corollary (26.11):
let $G$ be generic for Levy collapse $P$ and $X$ countable set of Ordinals in $M[G]$. Then there exists a $V[X]$-generic $...
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In Solovay's proof that generic extension is also generic over any intermediate model, why is transfinite induction required?
I'm trying to understand the proof of Lemma 4.4 in Solovay's famous paper "A Model of Set-Theory in Which Every Set of Reals is Lebesgue Measurable".
The lemma is slightly more complicated, ...
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Factor lemma for general iterated forcing
I am trying to write down by myself the proof of factor lemma, after staring at the proofs in Jech's textbook and Baumgartner's forcing survey and failed to gain much. It says an iteration forcing $\...
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Foundations of Forcing in Kunen
In Kunen's Set Theory book, forcing is described as a finitistic procedure to get some relative consistency result $Con(ZFC)\Rightarrow Con(T)$, where $T$ is an extension of $ZFC$. Because we can't ...
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When forcing, do we always have to have a generic filter on the forcing notion in $V$? What happens if we don't have one?
Suppose we force with a partial order $\mathbb{P}$ which lives in a model of set theory $M$. It's not too hard to show that if $M$ is countable, then there always is a filter $G$ in $V$ which is $\...
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Which forcing axioms may be destroyed through forcings which preserve $2^{\aleph_0}=\aleph_2$?
Assume ZFC; and the forcing axioms are referring to $\mathrm{MA}(\aleph_1)+2^{\aleph_0}=\aleph_2$, PFA, MM, or their variants.
I wonder if any of the axioms may be destroyed by forcing. If the forcing ...
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Random forcing does not add Cohen real.
I know that random forcing does not add unbounded reals. Is a Cohen real always unbounded?
I can proof that the product of two random forcings adds a Cohen real. How can I show that the two step ...
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Kunen problem on forcing $(2^{\aleph_0}=\aleph_2)^{\sf HOD}$ and $\sf L= HOD^{HOD}\subsetneq HOD\subsetneq V$
Exercise V.2.11, on p 320 of Kunen's Set Theory (the newer one) states
Starting with $\sf V=L$ in the ground model, follow the suggestion of
Exercise II.8.12 and obtain a generic extension satisfying ...
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Prove that if $c.c.(\mathbb{P}) < \omega$, then every $G$ that is $\mathbb{P}$-generic over $M$ is also in $M$
I'm having trouble with an exercise from Kunen's Introduction to Independence Proofs. This is Exercise F2 from Ch VII.
If $\mathbb{P} \in M$ and $c.c.(\mathbb{P}) < \omega$, show that every
$G$ ...
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Coding a new real using a variant of the Baumgartner forcing
I'm working with the following variant of Baumgartner's forcing to add a club subset of $\omega_1$, talked about by Mitchell on page 3 here: https://arxiv.org/pdf/math/0407225.pdf
Let $S \subseteq \...
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Henkin's Model Existence Theorem and Forcing
Henkin's Model Existence Theorem says that, if $T$ is a consistent theory, then there exists some set $S$ such that $S \models T$.
Suppose we are trying to prove the theorem $\text{Con}(ZFC) \...
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A Sacks-like forcing
Given a tree $T\subseteq \omega^{<\omega}$ we call T superperfect if
For all $t \in T$ there is an $s\supseteq t$ such that $s^\smallfrown n \in T$ for all $n\in\omega$, where $s^\smallfrown n$ is ...
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What is the definition of a Cohen real?
When I study set theory research, I sometimes come across "Cohen reals"(such as Joel David Hamkins' paper). However, I can't have found the definition of Cohen reals. What I know about Cohen ...
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Applications of Cohen reals (obtained by forcing) and maths without Continuum Hypothesis?
At it is well know, the von Neumann universe (or von Neumann hierarchy of sets) $V$ is a class of hereditary well-founded sets which can be used as a model for Zermelo–Fraenkel set theory (ZFC). This ...
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Can a Suslin tree squared collapse $\aleph_1$?
A standard fact is that if $T$ is a Suslin tree, then $T\times T$ flipped upside down and viewed as a forcing is not ccc. But can it collapse cardinals? At the very least, the answer is not always - ...
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Is forcing with open intervals on [0, 1] equivalent to Cohen forcing?
When I first tried to understand random forcing (which according to Wikipedia is forcing with $(\operatorname{Bor}(I),\subseteq,I)$, where $I = [0,1]$ and $\operatorname{Bor}(I)$ is the collection of ...
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Adding new bijections without adding new subsets by forcing
I found this post very interesting. It shows that, by forcing, any given cardinal $\kappa$ of the ground model can be made countable in a forcing extension by adding a bijection between $\omega$ and $\...
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Strange consistency proof based on countability
Let $\sigma(x)$ be a formula (in one free variable $x$) in the language of set theory such that $\mathsf{ZFC} \models \forall x (x \text{ countable } \Rightarrow \sigma(x))$. (E.g., $\sigma(x) =$ &...
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Prikry sequence is countable
I am reading the chapter on Prikry forcing of Handbook of Set Theory. On page 1353:
1.4 Lemma Let $G\subseteq \mathcal P$ be generic for $\langle \mathcal P, \le \rangle$. Then $\bigcup\{p\mid \...
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Is the Map $i$ a Complete or Dense Embedding from Prikry Silver forcing $\mathbb P\mathbb S$ to Sacksforcing $\mathbb S$?
Is the following map $i$ a complete embedding or a dense embedding from the Prikry-Silver forcing $\mathbb P\mathbb S$ to the Sacks forcing $\mathbb S$?
\begin{align}
i \colon \mathbb P\mathbb S \to \...
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Why can’t we use an arbitrary poset/Boolean algebra when doing set-theoretic forcing
I’m trying to read up on the Boolean-algebra approach to forcing, and I’m noticing a confusion that makes me thing I’ve misunderstood something fundamental about the process. Concretely, it seems to ...
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Naive way to violate SCH at $\aleph_\omega$
Silver proved that GCH can fail at a measurable cardinal; using Prikry forcing we get the failure of SCH at a singular cardinal of cofinality $\omega$. It seems that to bring that down to $\aleph_\...
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On complete embeddings between forcing notions
I was wondering whether the following two statement are true and why:
For any two forcing notions $\mathbb{P,Q}$, if $\ \Vdash_\mathbb{Q} ``$There is a $\check{\mathbb{P}}$-generic filter over $\dot{...
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In forcing, is Scott's trick "needed" to prove the axiom schema of replacement for M[G]?
I am studying (unramified) forcing, and am trying to prove the axiom schema of replacement for $M[G]$. To me, the idea seems to be vaguely similar to the proof that forcing with a c.c.c. poset doesn't ...
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Preserving powersets and preserving MA
I am reading a proof that seems to imply that if we start with a model of $\mathfrak{c} = \aleph_2 $ and $MA_{\aleph_1} $, and if a forcing preserves $\mathcal{P}(\omega_1)$, that is, it doesn't add ...
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Collapse $\aleph_{\omega_1}$ to $\aleph_\omega$.
Is there a forcing notion that collapses $\aleph_{\omega_1}$ to $\aleph_\omega$ while preserving every cardinal below $\aleph_\omega$?
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Omega-distributivity of the club shooting poset
I'm currently working through Lemma $23.9$ in Jech, which states (more or less) the following:
In some ground model $M$, let $S\subseteq \omega_1$ be stationary. Consider the notion of forcing $P_S$ ...
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Elementary substructure proof that kappa-cc forcing preserves stationary sets
Suppose $\kappa$ is a regular uncountable cardinal in the ground model $M$, and let $\mathbb P\in M$ be a forcing notion that has the $\kappa$ chain condition in $M$. Further suppose that $S\subseteq \...
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Borel codes and forcing
Can someone recommend to me some papers or books to learn about coding Borel sets and its absoluteness between transitive models (specially such generated by forcing)?
I was reading a paper about ...
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Is the Forcing Technique an additional independent axiom to ZFC?
Can the Forcing Technique introduced by Cohen be considered to be an axiom or is it a 'technique' with no additional assumptions to ZFC. So does Forcing introduce new objects that are not in V ? The ...
user239186
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Poset where no infinite descending chain has a lower bound
Suppose $(\mathbb{P},\leq)$ is a partial order that is (i) separative, i.e. if $x\nleq y$ then $\exists ~z\leq x$ s.t. no $w$ satisfies $w\leq z$ and $w\leq y$ (ii) every strictly descending chain ...
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Intuition behind the definition of "Support" in Forcing
I am reading "Characterization of generic extensions of models of set theory" in https://cmuc.karlin.mff.cuni.cz/pdf/cmuc1703/bukovsky.pdf.
Definition 1: For a relation r, let $ r''a \iff $ ...
user239186